Optimal. Leaf size=73 \[ -\frac {\cot ^7(c+d x)}{7 a^4 d}+\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {\cot (c+d x)}{a^4 d}+\frac {x}{a^4} \]
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Rubi [A] time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4120, 3473, 8} \[ -\frac {\cot ^7(c+d x)}{7 a^4 d}+\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {\cot (c+d x)}{a^4 d}+\frac {x}{a^4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 4120
Rubi steps
\begin {align*} \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx &=\frac {\int \cot ^8(c+d x) \, dx}{a^4}\\ &=-\frac {\cot ^7(c+d x)}{7 a^4 d}-\frac {\int \cot ^6(c+d x) \, dx}{a^4}\\ &=\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {\cot ^7(c+d x)}{7 a^4 d}+\frac {\int \cot ^4(c+d x) \, dx}{a^4}\\ &=-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {\cot ^7(c+d x)}{7 a^4 d}-\frac {\int \cot ^2(c+d x) \, dx}{a^4}\\ &=\frac {\cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {\cot ^7(c+d x)}{7 a^4 d}+\frac {\int 1 \, dx}{a^4}\\ &=\frac {x}{a^4}+\frac {\cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {\cot ^7(c+d x)}{7 a^4 d}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 36, normalized size = 0.49 \[ -\frac {\cot ^7(c+d x) \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};-\tan ^2(c+d x)\right )}{7 a^4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 147, normalized size = 2.01 \[ \frac {176 \, \cos \left (d x + c\right )^{7} - 406 \, \cos \left (d x + c\right )^{5} + 350 \, \cos \left (d x + c\right )^{3} + 105 \, {\left (d x \cos \left (d x + c\right )^{6} - 3 \, d x \cos \left (d x + c\right )^{4} + 3 \, d x \cos \left (d x + c\right )^{2} - d x\right )} \sin \left (d x + c\right ) - 105 \, \cos \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 139, normalized size = 1.90 \[ \frac {\frac {13440 \, {\left (d x + c\right )}}{a^{4}} + \frac {9765 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1295 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} + \frac {15 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1295 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9765 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 79, normalized size = 1.08 \[ -\frac {1}{7 d \,a^{4} \tan \left (d x +c \right )^{7}}-\frac {1}{3 d \,a^{4} \tan \left (d x +c \right )^{3}}+\frac {1}{5 d \,a^{4} \tan \left (d x +c \right )^{5}}+\frac {1}{d \,a^{4} \tan \left (d x +c \right )}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 60, normalized size = 0.82 \[ \frac {\frac {105 \, {\left (d x + c\right )}}{a^{4}} + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{a^{4} \tan \left (d x + c\right )^{7}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.93, size = 51, normalized size = 0.70 \[ \frac {x}{a^4}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^6-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{3}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{5}-\frac {1}{7}}{a^4\,d\,{\mathrm {tan}\left (c+d\,x\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{\sec ^{8}{\left (c + d x \right )} - 4 \sec ^{6}{\left (c + d x \right )} + 6 \sec ^{4}{\left (c + d x \right )} - 4 \sec ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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